An Improved Grey Wolf Optimizer and Its Application in Robot Path Planning

This paper discusses a hybrid grey wolf optimizer utilizing a clone selection algorithm (pGWO-CSA) to overcome the disadvantages of a standard grey wolf optimizer (GWO), such as slow convergence speed, low accuracy in the single-peak function, and easily falling into local optimum in the multi-peak function and complex problems. The modifications of the proposed pGWO-CSA could be classified into the following three aspects. Firstly, a nonlinear function is used instead of a linear function for adjusting the iterative attenuation of the convergence factor to balance exploitation and exploration automatically. Then, an optimal α wolf is designed which will not be affected by the wolves β and δ with poor fitness in the position updating strategy; the second-best β wolf is designed, which will be affected by the low fitness value of the δ wolf. Finally, the cloning and super-mutation of the clonal selection algorithm (CSA) are introduced into GWO to enhance the ability to jump out of the local optimum. In the experimental part, 15 benchmark functions are selected to perform the function optimization tasks to reveal the performance of pGWO-CSA further. Due to the statistical analysis of the obtained experimental data, the pGWO-CSA is superior to these classical swarm intelligence algorithms, GWO, and related variants. Furthermore, in order to verify the applicability of the algorithm, it was applied to the robot path-planning problem and obtained excellent results.


Introduction
The metaheuristic algorithm is an improvement of the heuristic algorithm combined with a random algorithm and local search algorithm to implement optimization tasks. In recent years, metaheuristic optimization has made some recent developments. Jiang X et al. proposed optimal pathfinding with a beetle antennae search algorithm using ant colony optimization initialization and different searching strategies [1]. Khan A H et al. proposed BAS-ADAM: an ADAM-based approach to improving the performance of beetle antennae search optimizer [2]. Ye et al. proposed a modified multi-objective cuckoo search mechanism and applied this algorithm to the obstacle avoidance problem of multiple uncrewed aerial vehicles (UAVs) for seeking a safe route by optimizing the coordinated formation control of UAVs to ensure the horizontal airspeed, yaw angle, altitude, and altitude rate are converged to the expected level within a given time for inverse kinematics and optimization [3]. Khan et al. proposed using the social behavior of beetles to establish a computational model for operational management [4]. As one of the latest metaheuristic algorithms, grey wolf optimizer (GWO) is widely employed to settle real industrial issues because GWO maintains a balance between exploitation and exploration through dynamic parameters and has a strong ability to explore the rugged search space of the problem [5,6], such as the selection problem [7][8][9], privacy protection issue [10], adaptive weight problem [11], smart home scheduling problem [12], prediction problem [13][14][15], classification problem [16], and optimization problem [17][18][19][20].
(1) Surround the prey In the process of hunting, in order to surround the prey, it is necessary to calculate the distance between the current grey wolf and the prey and then update the position according to the distance. The behavior of grey wolves rounding up prey is defined as follows: and where Formula (1) is the updating formula of the grey wolf's position, and Formula (2) is the calculation formula of the distance between the grey wolf individual and prey. Variable t is the current iteration number, X P (t) and X(t) are the current position vectors of the prey and the grey wolf at iteration t, respectively. A and C are coefficient vectors calculated by Formula (3) and Formula (4), respectively. and where a is the convergence factor, and a linearly decreases from 2 to 0 as the number of iterations increases. r 1 and r 2 are random vectors in [0, 1]. Formula (5) is the calculation formula a and t max indicates the maximum number of iterations.
(2) Hunting In an abstract search space, the position of the optimal solution is uncertain. In order to simulate the hunting behavior of grey wolves, α, β, and δ wolves are assumed to have a better understanding of the potential location of prey. α wolf is regarded as the optimal solution, β wolf is regarded as the suboptimal solution, and δ wolf is regarded as the third and where D α represents the distance between the current grey wolf and α wolf; D β represents the distance between the current grey wolf and β wolf; D δ represents the distance between the current grey wolf and δ wolf; and X α , X β , and X δ represent the position vectors of α wolf, β wolf, and δ wolf, respectively. X(t) is the current position of the grey wolf. C 1 , C 2 , and C 3 are random vectors, calculated by Formula (4). A 1 , A 2 , and A 3 are determined by Formula (3). Formula (7) represents the step length and direction of grey wolf individuals to α, β, and δ wolves, and Formula (8) is the position-updating formula of grey wolf individuals. According to the description above, the algorithm flow chart of GWO is shown in Figure 1.

CSA
The CSA was proposed by De Castro and Von Zuben in 2002 according to the clonal selection theory [32]. The CSA simulates the mechanism of immunological multiplication, mutation, and selection and is widely used in many problems.
For the convenience of model design, the principle of the biological immune system is simplified. All substances that do not belong to themselves are regarded as antigens.

CSA
The CSA was proposed by De Castro and Von Zuben in 2002 according to the clonal selection theory [32]. The CSA simulates the mechanism of immunological multiplication, mutation, and selection and is widely used in many problems.
For the convenience of model design, the principle of the biological immune system is simplified. All substances that do not belong to themselves are regarded as antigens. When the immune system is stimulated by antigens, antibodies will be produced to bind to antigens specifically. The stronger the association between antigen and antibody, the higher the affinity. Then, the antibodies with high antigen affinity are selected to multiply and differentiate between binding to the antigens, increase their antigen affinity through super-mutation, and finally eliminate the antigens. In addition, some of the antibodies are converted into memory cells in order to respond quickly to the same or similar antigens in the future. In the CSA, the problem that needs to be solved is regarded as the antigen, and the solution to the problem is regarded as the antibody. At the same time, the receptorediting mechanism is adopted to avoid falling into the local optimum. The flow chart of CSA is shown in Figure 2. The above is the introduction of the GWO and CSA. The proposed algorithm in this paper is also inspired by GWO and CSA. The pGWO-CSA proposed in this paper is introduced in detail in Section 3 below.  The above is the introduction of the GWO and CSA. The proposed algorithm in this paper is also inspired by GWO and CSA. The pGWO-CSA proposed in this paper is introduced in detail in Section 3 below.

The Proposed pGWO-CSA
In order to improve the convergence speed and accuracy in the single-peak function and the ability to jump out of local optimum in the multi-peak function and complex problems: Firstly, a nonlinear function is used instead of a linear function for adjusting the iterative attenuation of convergence factor to balance exploitation and exploration automatically; Secondly, the pGWO-CSA adopts a new position-updating strategy, and different position-updating strategies are used for α wolf, β wolf, and other wolves, so that the position updating of α wolf and β wolf are not affected by the wolves with lower fitness; Finally, the pGWO-CSA combines GWO with CSA and introduce the cloning and super-mutation of the CSA into GWO.
The detailed improvement strategy is as follows.

Replace Linear Function with Nonlinear Function
In GWO, a decreases from 2 to 0 as the number of iterations increases, and the range of A decreases as a decreases. According to Formulas (6) and (7), when |A| < 1, the next position of the grey wolf can be anywhere between the current position and the prey, and the grey wolf approaches the prey guided by α, β, and δ. When |A| ≥ 1, the grey wolf moves away from the current α, β, and δ wolves and searches for the optimal global value. Therefore, when |A| < 1, grey wolves approach their prey for exploitation. When |A| ≥ 1, grey wolves move away from their prey for exploration. In the original GWO, the parameter a linearly decreases from 2 to 0, with half of the iterations devoted to exploitation and half to exploration. In order to balance exploitation and exploration, the pGWO-CSA adopts a nonlinear function instead of a linear function to adjust the iterative attenuation of parameter a so as to enhance the exploration ability of the grey wolf at the early stage of iteration. In pGWO-CSA, parameter a is calculated by Formula (9).
where variable t is the current iteration number, t max is the maximum iteration number, and u is the coefficient, where the value in this paper is 2.
Iterative curves of parameter a in the original GWO and pGWO-CSA are shown in Figure 3.

The Proposed pGWO-CSA
In order to improve the convergence speed and accuracy in the single-peak function and the ability to jump out of local optimum in the multi-peak function and complex problems: Firstly, a nonlinear function is used instead of a linear function for adjusting the iterative attenuation of convergence factor to balance exploitation and exploration automatically; Secondly, the pGWO-CSA adopts a new position-updating strategy, and different position-updating strategies are used for α wolf, β wolf, and other wolves, so that the position updating of α wolf and β wolf are not affected by the wolves with lower fitness; Finally, the pGWO-CSA combines GWO with CSA and introduce the cloning and supermutation of the CSA into GWO.
The detailed improvement strategy is as follows.

Replace Linear Function with Nonlinear Function
In GWO, a decreases from 2 to 0 as the number of iterations increases, and the range of A decreases as a decreases. According to Formulas (6) and (7), when 1 A < , the next position of the grey wolf can be anywhere between the current position and the prey, and the grey wolf approaches the prey guided by α, β, and δ. When 1 A ≥ , the grey wolf moves away from the current α, β, and δ wolves and searches for the optimal global value. Therefore, when 1 A < , grey wolves approach their prey for exploitation. When 1 A ≥ , grey wolves move away from their prey for exploration. In the original GWO, the parameter a linearly decreases from 2 to 0, with half of the iterations devoted to exploitation and half to exploration. In order to balance exploitation and exploration, the pGWO-CSA adopts a nonlinear function instead of a linear function to adjust the iterative attenuation of parameter a so as to enhance the exploration ability of the grey wolf at the early stage of iteration. In pGWO-CSA, parameter a is calculated by Formula (9).
where variable t is the current iteration number, is the maximum iteration number, and u is the coefficient, where the value in this paper is 2.
Iterative curves of parameter a in the original GWO and pGWO-CSA are shown in Figure 3.  As can be seen from Figure 3, the convergence factor slowly decays in the early stage, improving the global search ability, and rapidly decays in the later stage, accelerating the search speed and optimizing the global exploration and local development ability of the algorithm.

Improve the Grey Wolf Position Updating Strategy
In the original grey wolf algorithm, the positions of all grey wolves in each iteration are updated by Formulas (6)- (8). In the position-updating strategy, the position updating of the α wolf is affected by the β wolf and δ wolf with poor fitness. The position updating of the β wolf is affected by the δ wolf with poor fitness.
Therefore, a new location updating strategy is proposed in this paper. In each iteration, the fitness of grey wolves is calculated, and the top three wolves α, β, and δ with the best fitness are saved and recorded. The specific location update formula is as follows.
where X 1 , X 2 , and X 3 are determined by Formula (7). X (t) represents the pre-update position. On this basis, if the current α wolf and β wolf are close to the optimal solution, α wolf and β wolf have a greater probability to update to the position with better fitness so as to better guide wolves to hunt the prey and find the optimal solution. If α wolf and β wolf are in the local optimum, other wolves still update their positions according to Formulas (6)-(8) so that algorithm will not fall into the local optimum. Therefore, the proposed improved method can not only improve the exploitation capability but also not affect the exploration capability.

Combine GWO with CSA
GWO is combined with CSA by introducing the cloning and super-mutation of the CSA into GWO, and the exploitation and exploration ability of GWO is improved. For each grey wolf, a super-mutation coefficient (Sc) and a random number (r3) are introduced. The wolf with good adaptability has a small coefficient of super-variance and a small probability of variation, while the wolf with poor adaptability has a large probability of variation. If the super-variation coefficient Sc of the current grey wolf is greater than the random number r3, the current grey wolf will be cloned, and then the cloned grey wolf will be mutated through Formulas (6)- (8). If the mutated grey wolf has comparatively better adaptability, it will replace the current grey wolf. The specific calculation formula is as follows. and where f itness i represents the fitness of the current grey wolf, f itness min represents the fitness of the best wolf, and f itness max represents the fitness of the worst wolf. r3 is a random number between [0, 1]. X (t) is determined by Formula (10). X (t) represents the best between X (t) and X (t) as the result of the variation of X(t) through Formulas (6)-(8).

Algorithm Flow Chart of pGWO-CSA
According to the improvement idea mentioned above, the algorithm flow chart of pGWO-CSA is shown in Figure 4.

Algorithm Flow Chart of pGWO-CSA
According to the improvement idea mentioned above, the algorithm flow chart of pGWO-CSA is shown in Figure 4.

Time Complexity Analysis of the Algorithm
Assuming that the population size is N, the dimension of objective function F is Dim, and the number of iterations is T, the time complexity of the pGWO-CSA algorithm can be calculated as follows.
First, the time complexity required to initialize the grey wolf population is O(N × Dim), the time complexity required to calculate the fitness of all grey wolves is O(N × F (Dim)), and the time complexity required to preserve the location of the best three wolves is O(3 × Dim).
Then, in each iteration, the time complexity required to complete all grey wolves' position updating is O(N × Dim), the time complexity required to update a, A, and C is O(1), and the time complexity required to calculate the fitness of all grey wolves is O(N × F(Dim)). The time complexity of cloning and super-mutation is O(N1 × Dim), where N1 is

Time Complexity Analysis of the Algorithm
Assuming that the population size is N, the dimension of objective function F is Dim, and the number of iterations is T, the time complexity of the pGWO-CSA algorithm can be calculated as follows.
First, the time complexity required to initialize the grey wolf population is O(N × Dim), the time complexity required to calculate the fitness of all grey wolves is O(N × F (Dim)), and the time complexity required to preserve the location of the best three wolves is O(3 × Dim).
Then, in each iteration, the time complexity required to complete all grey wolves' position updating is O(N × Dim), the time complexity required to update a, A, and C is O(1), and the time complexity required to calculate the fitness of all grey wolves is O(N × F(Dim)). The time complexity of cloning and super-mutation is O(N1 × Dim), where N1 is the number of wolves meeting the mutating condition, and the time complexity of updating the fitness and location of α, β, and δ is O(3 × Dim). The total iteration is T times. So, the total time complexity is O( So, in the worst case, the time complexity of the whole algorithm is O(

Experimental Test
In this section, 15 benchmark functions from F1-F15 are selected to test the performance of pGWO-CSA. Firstly, pGWO-CSA is compared with other swarm intelligence algorithms. Then pGWO-CSA is compared with GWO and its variants. Table 1 describes these benchmark functions in detail. Section 4.1 will compare pGWO-CSA with other swarm intelligence algorithms. Section 4.2 will compare pGWO-CSA with GWO and its variants.

No.
Function Among these benchmark functions, the first seven benchmark functions, F1-F7, are simpler, while the last eight benchmark functions, F8-F15, are more complex. The dimension of these benchmark functions is 30 dimensions and 50 dimensions. The population size of all algorithms is set to 30, the maximal iteration of all algorithms is set to 500, and all experimental data are measured on the same computer to ensure a fair comparison between different algorithms. In order to avoid the randomness of the algorithm, each algorithm in this paper will be run on each test function 30 times. At the same time, the mean, standard deviation, and minimum and maximum values of the running results are recorded.

Compare with Other Swarm Intelligence Algorithms
In the comparison between pGWO-CSA and other swarm intelligence algorithms, PSO [29], DE [30], and FA [31] are selected to compare with pGWO-CSA. For each algorithm, the function optimization task is performed on the test functions F1-F15 in 30 dimensions and 50 dimensions, and the mean, standard deviation, and minimum and maximum values of the running results are recorded. The main parameters of the PSO, DE, FA, and pGWO-CSA are shown in Table 2. Table 2. Main parameters of the four algorithms.

Algorithm
The Main Parameters In 30 dimensions, the test results of these four algorithms on test function F1-F15 are shown in Table 3. The convergence curves of these four algorithms on test function F1-F15 are recorded in Figures 5a, 6a, 7a, 8a, 9a, 10a, 11a, 12a, 13a, 14a, 15a, 16a, 17a, 18a and 19a. in 14 out of 15 test functions, and DE outperformed pGWO-CSA only in test function F6. Compared with FA, pGWO-CSA outperformed FA in all 15 test functions.
In terms of the performance of the standard deviation. Sort by the number of optimal values. The pGWO-CSA ranked first with 11 optimal values. PSO ranked second with three optimal values. FA ranked third with one optimal value. DE ranked fourth with zero optimal values. Compared with PSO, pGWO-CSA outperformed PSO in 11  In addition, the pGWO-CSA can find theoretical optimal values on the test functions F9, F11, and F12. In the test functions F1, F2, F3, F4, F10, F11, F13, and F15, pGWO-CSA is superior to PSO, DE, and FA in terms of the mean, standard deviation, and minimum and maximum. Although PSO outperformed pGWO-CSA in the mean, standard deviation, and minimum and maximum on test function F14, pGWO-CSA still outperformed DE and FA on test function F14.                                              In 50 dimensions, the test results of these four algorithms on test function F1-F15 are shown in Table 4. The convergence curves of these four algorithms on test function F1-F15 are recorded in Figures 5b-19b. In 50 dimensions, the test results of these four algorithms on test function F1-F15 are shown in Table 4. The convergence curves of these four algorithms on test function F1-F15 are recorded in Figures 5b, 6b, 7b, 8b, 9b, 10b, 11b, 12b, 13b, 14b, 15b, 16b, 17b, 18b and 19b.  In addition, pGWO-CSA can find theoretical optimal values on the test functions F9, F11, and F12. In the test functions F1, F2, F3, F4, F9, F10, F13, and F15, pGWO-CSA is superior to PSO, DE, and FA in terms of the mean, standard deviation, and minimum and maximum. Although pGWO-CSA is not as good as FA on test function F6 and as good as PSO on test function F14, pGWO-CSA still outperformed the other two swarm intelligence algorithms on these two functions.
Based on the above data and analysis, pGWO-CSA has faster convergence speed, higher accuracy, and better ability to jump out of local optimum compared with other swarm intelligence algorithms in either 30 or 50 dimensions. In order to further verify the performance of pGWO-CSA, we will next compare pGWO-CSA with GWO and its variants.

Compare with GWO and Its Variants
In order to further verify the performance of pGWO-CSA, pGWO-CSA is compared with GWO [5] and its variants OGWO [27], DGWO1, and DGWO2 [28] on the test functions F1-F15 in 30 dimensions and 50 dimensions. The main parameters of pGWO-CSA, GWO, OGWO, DGWO1, and DGWO2 are shown in Table 5. Table 5. Main parameters of the five algorithms.
In terms of the performance of the mean. Sort by the number of optimal values. The pGWO-CSA ranked first with eight optimal values. DGWO2 ranked second with six optimal values. OGWO ranked third with three optimal values. DGWO1 and GWO tied for fourth place with one optimal value. Compared with GWO, pGWO-CSA outperformed GWO in 14  In terms of the performance of the standard deviation. Sort by the number of optimal values. The pGWO-CSA ranked first with eight optimal values. DGWO2 ranked second with seven optimal values. OGWO ranked third with two optimal values. GWO ranked fourth with one optimal value. DGWO1 ranked fifth with zero optimal values. Compared with GWO, pGWO-CSA outperformed GWO in 13  In terms of the performance of the maximum. Sort by the number of optimal values. DGWO2 ranked first with eight optimal values. The pGWO-CSA ranked second with seven optimal values. OGWO ranked third with three optimal values. GWO ranked fourth with one optimal value. DGWO1 ranked fifth with zero optimal values. Compared with GWO, pGWO-CSA outperformed GWO in 12  In addition, pGWO-CSA can find theoretical optimal values on the test functions F9, F11, and F12. In the test functions F8, F9, F11, and F12, pGWO-CSA is the optimal value among the five algorithms in terms of the mean, standard deviation, and minimum and maximum. It is not difficult to find from Tables 6 and 7 that DGWO2 performs better than pGWO-CSA in the first four test functions, F1-F4, in both the 30 dimensions and the 50 dimensions. As can be seen from Table 1, the first four test functions are simple single-peak functions, indicating that DGWO1 performs better than pGWO-CSA in simple single-peak functions. However, compared with the other three algorithms, pGWO-CSA still performs better on the test functions F1-F4.
By comparison, it is not difficult to find that pGWO-CSA performs better than the previous seven test functions, F1-F7, in the following eight test functions, F8-F15, whether in the 30 dimensions or the 50 dimensions. It can be seen that pGWO-CSA performs better in more complex functions, which is largely due to the super-mutation operation carried out by pGWO-CSA, which helps pGWO-CSA better jump out of local optimum.
Based on the above data and analysis, pGWO-CSA has faster convergence speed, higher accuracy, and better ability to jump out of the local optimum compared with GWO and its variants in either 30 or 50 dimensions. In order to further reveal the performance of pGWO-CSA, the Wilcoxon test is performed in Section 4.3 based on the experimental data in Sections 4.1 and 4.2.

Wilcoxon Test
In order to further reveal the performance of pGWO-CSA, according to the experimental data in Sections 4.1 and 4.2, the Wilcoxon test is conducted on the mean of the 30 running results of each algorithm. The statistical results are shown in Table 8. In the Wilcoxon test, '+' means that the proposed algorithm is inferior to the selected algorithm, '−' means that the proposed algorithm is superior to the selected algorithm, and '=' means that the two algorithms get the same result.  28  29  29  28  24  28  18  =  0  0  0  2  2  2  2 It can be seen from Table 8 that the number of '+' of each algorithm is small, indicating that the seven algorithms compared with pGWO-CSA only outperform pGWO-CSA in a few test functions, and the number of '−' of each algorithm exceeds 15, indicating that pGWO-CSA outperformed other algorithms in most test functions. The results show that pGWO-CSA is superior to other swarm intelligence algorithms, GWO, and its variants.

Robot Path-Planning Problem
With the development of artificial intelligence, robots have been widely used in various fields [33][34][35]. Among them, robot path planning is an important research problem. To further verify the applicability and superiority of the proposed algorithm, it is applied to the robot path-planning problem.

Robot Path-Planning Problem Description
The robot path-planning problem mainly includes two aspects: environment modeling and evaluation function. Environment modeling is to transform the environmental information of the robot into a form that can be recognized and expressed by a computer. The evaluation function is used to measure the path quality and is regarded as the objective function to be optimized by the algorithm.

Environment Modeling
The environment model of the robot path-planning problem is shown in Figure 35. The starting point is located at (0,0) and marked with a black star, the endpoint is located at (10,10) and marked with a blue triangle, and the obstacles are marked with a green circle. The mathematical expression of the obstacles is shown in Formula (13).
where a and b represent the center coordinates of the obstacle, and r represents the radius of the circle.

Evaluation Function
Suppose the robot finds some path points from start to end: ( , ), ( , ), ..., ( , ), and the coordinate of the path is ( , ). A complete path formed by connecting these path points is a feasible solution to the robot path-planning problem. In order to

Evaluation Function
Suppose the robot finds some path points from start to end: (x 0 , y 0 ), (x 1 , y 1 ), . . . , (x n , y n ), and the coordinate of the path point i is (x i , y i ). A complete path formed by connecting these path points is a feasible solution to the robot path-planning problem. In order to reduce the optimization dimension of the problem and smooth the path curve, the spline interpolation method is used to construct the path curve. In order to evaluate the quality of the path, this paper considers the length of the path and the risk of the path. The evaluation function is shown in Formula (14).
where w1 and w2 are weight parameters and w1 + w2 = 1.0. f it len represents the fitness value of the length of the path, which is calculated by Formula (15); f it rick represents the fitness value of the risk of the path, which is calculated by Formula (16).
where n is the total number of path points, and (x i , y i ) represents the coordinate of the path point i .
where c is the penalty coefficient, k is the total number of obstacles, (a i , b i ) is the coordinates of the center of obstacle i, and r i is the radius. According to Formulas (14)- (16), when the fitness value of f it len is small, then the length of the path is short. When the fitness value of f it risk is small, the risk of the path is low. Therefore, the smaller the f it, the higher the quality of the path.

The Experimental Results
In order to verify the applicability and superiority of pGWO-CSA in robot pathplanning problems, PSO, DE, FA, GWO, and its variants are compared with pGWO-CSA. The parameters of all algorithms are exactly the same as in Section 4. In order to avoid the randomness of the algorithm, each algorithm will be run 10 times, and then the minimum, maximum, and mean of the results will be recorded. The path planned by pGWO-CSA is shown in Figure 36, and the experimental results of all algorithms are shown in Table 9. According to the experimental data, pGWO-CSA is the optimal value of all algorithms in the performance of the minimum, maximum, and mean. The applicability and superiority of pGWO-CSA are further verified.
parameters of all algorithms are exactly the same as in Section 4. In order to avoid the randomness of the algorithm, each algorithm will be run 10 times, and then the minimum, maximum, and mean of the results will be recorded. The path planned by pGWO-CSA is shown in Figure 36, and the experimental results of all algorithms are shown in Table 9. According to the experimental data, pGWO-CSA is the optimal value of all algorithms in the performance of the minimum, maximum, and mean. The applicability and superiority of pGWO-CSA are further verified.

Conclusions
Aiming at the defects of the GWO, such as low convergence accuracy and easy precocity when dealing with complex problems, this paper proposes pGWO-CSA to settle these drawbacks. Firstly, the pGWO-CSA uses a nonlinear function instead of a linear function to adjust the iterative attenuation of the convergence factor to balance exploitation and exploration. Secondly, pGWO-CSA improves GWO's position-updating strategy, and finally, pGWO-CSA is mixed with the CSA. The improved pGWO-CSA improves the convergence speed, precision, and ability to jump out of the local optimum. The experimental results show that the pGWO-CSA has obvious accuracy advantages. Compared with GWO and its variants participating in the experiment, the pGWO-CSA shows good stability in both 30 and 50 dimensions and is suitable for the optimization of complex and variable problems. Finally, the proposed algorithm is applied to the robot path-planning problem, which further verifies the applicability and superiority of the proposed algorithm.

Conflicts of Interest:
The authors declare no conflict of interest.